As far as I know, if two angles are congruent, they are equal. But if two triangles are congruent, they are not necessarily equal. If they are equal, they must be congruent however. The definition here would be, equal in relative dimensions but different in scale.

Metamathematically speaking, "equality" is usually related to the concept of "identity". If you're given two different angles, you can tell which angle is which, so they cannot be equal angles. (Even if they have the same angle measure!)

Two angles ABC and DFE, say, are "equal" if and only if there are exactly the same angle: that would mean that B and F are just different ways of referring to the same point and that A is on one of the rays FD or FE and C is on the other one.

Two angles are "congruent" if and only if they have the same measure.

Generally speaking, in mathematics, we use the term "equal" to mean "(possibly different) ways of referring to exactly the same object" and "congruent" to mean "have some specific property in common".

As far as I know, if two angles are congruent, they are equal. But if two triangles are congruent, they are not necessarily equal. If they are equal, they must be congruent however. The definition here would be, equal in relative dimensions but different in scale.

This is not so. Yes, for a triangle to be equal it would have to be the same exact triangle (by the reflexive property or line ae is equal to line ea)
However, a triagngle that's "equal in relative dimensions but different in scale" would be similar NOT congruent.
Congruantcy in triangles, or polygons in general, is a special case of similarity.

Similarity is when all corresponding angles are congruent (equal in degree, you were right about that part) and all corresponding lengths are proportional by the same scale factor.

What's a scale factor? take a ratio of the corresponding sides of the two similar triangles (with the length of the side of the larger trinagle on top), treat it like a fraction and divide it out into a decimal, there's your scale factor. Take any length of the smaller triangle multiply it by this and you have the length of the corresponding side of the larger triangle. It is from this idea that cross multiplication was made, though how it's simpler I'll never know it dousn't elimnate any steps and it's much less logical and easy to figure out.

Anyway, back to the explanation. A congruent triangle is one where the triangle is similar (thus congruent angles and proportional sides) and the sides are proportional by a ratio of 1:1 (on toher words the lengths have the same value).

Thus, equal triangles are triangles which are exactly the same, as in trinagle BCE is equal to triangle ECB.
Similar triangles have congruent corresponding angles and proportional corresponding side lengths (whihc are all proportional by the same amount or scale factor).
And Congruent triangles are similar triangles wiht a scale factor of 1:1, or in other words congruent corresponding angles and congruent corresponding sides (ea=ae, reflexive property). In other words, the triangles have the same exct measurements in all ways but are not in exactly the same place and are not exactly the same triangle and thus are not equal.
You were right that triangles can be congruent but not equal, but an equal triangle must be congruent. Similrity dousn't neccassarily mean the special case of congruency not to mention the even more special case of equalness. Congruency dousn't neccassarily mean that it's the special case of equalness but it does mean that it HAS to be similar. Equalness in triangles means that it HAS to be not only similar but also congruent.

Halls of Ivy is correct about everything but, "Generally speaking, in mathematics, we use the term "equal" to mean "(possibly different) ways of referring to exactly the same object" and "congruent" to mean "have some specific property in common". "
Your definition of congruency applies for many usages however, in hte case of eometric shapes such as triangles this would only apply for similarity (you said "have some specific property in common" this property would be the angles, however as you said "have some" this menas the lengths don't necassarily have to be congruent. Of cource, in reality at least with triangles they would still have to be congruent as a triangle with certain corresponding angle measurements must have proportional sides as trinagles are rigid. I don't know as much about other shapes though so I don't know if this holds true with them, but I reason it logically probably does for reasons which are complicated and I won't get into. BTW a square dousn't count as a special exception since a square is a special form of rectangle.)